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- In quantum mechanics, the measurements of two different physical properties are represented by the operators A and B. It is possible to know, exactly and simultaneously, the values of both measured quantities only if the (a) eigenfunctions of operator A form an orthonormal set and the eigenfunctions of operator B form an orthonormal set. (b) eigenfunctions for both operators A and B can be normalized. (c) eigenvalues for both operators A and B are real numbers. (d) operators A and B are both Hermitian. (e) operators A and B commute.2. Commuting operators. (a) Show whether the operator of the kinetic energy operator K and the momentum operator Px for a one-dimensional system commute (show every step). (b) What does the result mean (physical interpretation)?(c) Evaluate the commutator for the operators (a) x³ and — (b) px and p3 dx d² and e-cx (d) and ax² + bx + c dx²
- Solve the following linear equations for x.(a) 7x + 5 =0(b)-4x +3 =0(c) -3x = -2Which quantum number determines the shape of an orbital? (I) schrodinger quantum number (ii) principal quantum number (iii) magnetic quantum number (iv)Angular momentum quantum number (V) Heisenberg Quantum numberConsider an electron in the N shell. (a) What is the smallest orbital angular momentum it could have? (b) What is the largest orbital angular momentum it could have? Express your answers in terms of h and in SI units. (c) What is the largest orbital momentum this electron could have in any chosen direction? Express your answers in terms of h and in SI units. (d) What is the largest spin angular momentum this electron could have in any chosen direction? Express your answers in terms of h and in SI units. (e) For the electron in part (c), what is the ratio of its spin angular momentum in the z-direction to its orbital angular momentum in the z direction? E and d no answer
- (1) (2) (b) Look very carefully at the picture below. Give the relevant quantum numbers. Explain your answer. y-axis (c) (1) What is a wavefunction? (ii) What are the two parts of a wavefunction?If an electron falls through an electric potential difference of one volt, it acquires an energy of one electron volt. a) If the electron is to have a wavelength of 0.1 nm, what potential difference must it pass through? b) What is the velocity of this electron?(QUANTUM CHEMISTRY) Dscribe how a wavefunction summarizes the dynamical properties of a system and how those properties may be predicted. Discuss the relation between probability amplitude, probability density, and probability. Describe the constraints that the Born interpretation puts on acceptable wavefunctions. What are the advantages of working with normalized wavefunctions?
- Write the normalized form of the ground state wavefunction of the harmonic oscillator in terms of the variable y and the parameter α. (a) Write the integral you would need to evaluate to find the mean displacement <y>, and then use a symmetry argument to explain why this integral is equal to 0. (b) Calculate <y2> (the necessary integral will be found in the Resource section). (c) Repeat the process for the first excited state.2. Consider a helium atom. (a) Sketch the system and write the Hamiltonian. Denote the term(s) in the operator that make(s) the Schrödinger equation unsolvable. (b) Given an excited-state configuration 1s 2s', express all possible excited-state wavefunctions in terms of spatial and spin functions for the two electrons. For each, denote the symmetry of the overall wavefunction, the spatial component, and the spin component with respect to exchange.Using the wavefunctions for the particle in 1D box, show that the Hamiltonian is (a) a linear operator, and (b) Hermitian.